RMS Value and Average Power
Find the DC that heats the same
Slide the amplitude of the sine wave and watch the RMS value. The peak reaches the amplitude, but in its heating effect on a resistor it equals a single DC value smaller than the peak. Set the amplitude so that RMS value reaches the target of 220 V.
Instantaneous power keeps sloshing
The instantaneous power into a resistor is the voltage squared over the resistance at each moment, v²/R. When the voltage passes zero the power is zero; at the peak the power is greatest. Because it is squared, the power is always positive even when the voltage is negative. To name this sloshing power with one value, you take its average over a full cycle.
The mean of the square is one half
The square of a sine v = V_peak sin swings between the peak V_peak² and 0, and its average over one cycle is exactly half of V_peak². So the average power is (V_peak²/2)/R. Wanting to write this like DC as P = V_rms²/R, we define V_rms² = V_peak²/2. That is, V_rms = V_peak/√2.
The RMS value is the equivalent DC
The real meaning of the RMS value is the DC voltage that delivers the same average power to the same resistor. The heating effect of 220 V AC on a bulb is identical to that of 220 V DC. So calling the outlet 220 V refers to this RMS value, not the peak (about 311 V). Once you switch to RMS values, AC power calculations become as simple as DC.
Back to the first screen
As you slid the amplitude, the RMS line tracked it always a factor of √2 below the peak, and RMS reached 220 V when the peak was about 311 V. For all its restless swinging, in its power to heat a resistor the waveform equaled a single 220 V DC. The one number called RMS translates the sloshing into DC, letting you handle AC power simply.